Closed convex function

{{Short description|Terms in Maths}} In mathematics, a function <math>f: \mathbb{R}^n \rightarrow \mathbb{R} </math> is said to be '''closed''' if for each <math> \alpha \in \mathbb{R}</math>, the sublevel set <math> \{ x \in \mbox{dom} f \vert f(x) \leq \alpha \} </math> is a closed set.

Equivalently, if the epigraph defined by <math> \mbox{epi} f = \{ (x,t) \in \mathbb{R}^{n+1} \vert x \in \mbox{dom} f,\; f(x) \leq t\} </math> is closed, then the function <math> f </math> is closed.

This definition is valid for any function, but most used for convex functions. A proper convex function is closed if and only if it is lower semi-continuous.<ref>{{Cite book|title = Convex Optimization Theory|publisher = Athena Scientific|year = 2009|isbn = 978-1886529311|pages=10, 11 }}</ref>

==Properties==

* If <math>f: \mathbb{R}^n \rightarrow \mathbb{R} </math> is a continuous function and <math>\mbox{dom} f </math> is closed, then <math> f</math> is closed. * If <math>f: \mathbb R^n \rightarrow \mathbb R </math> is a continuous function and <math>\mbox{dom} f </math> is open, then <math> f </math> is closed if and only if it converges to <math>\infty</math> along every sequence converging to a boundary point of <math>\mbox{dom} f </math>.<ref>{{cite book|last1=Boyd|first1=Stephen|first2=Lieven|last2=Vandenberghe|title=Convex optimization|date=2004|publisher=Cambridge|location=New York|isbn=978-0521833783|pages=639–640|url=https://www.stanford.edu/~boyd/cvxbook/bv_cvxbook.pdf}}</ref> * A closed proper convex function ''f'' is the pointwise supremum of the collection of all affine functions ''h'' such that ''h'' ≤ ''f'' (called the affine minorants of ''f'').

==References== {{Reflist}}

* {{cite book|author=Rockafellar, R. Tyrrell|author-link=Rockafellar, R. Tyrrell|title=Convex Analysis|publisher=Princeton University Press|location=Princeton, NJ|year=1997|orig-year=1970|isbn=978-0-691-01586-6}}

Category:Convex analysis Category:Types of functions